Table Of ContentHans-Heinrich Körle
Infinite Series in a History of Analysis
De Gruyter Textbook
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Hans-Heinrich Körle
Infinite Series in a
History of Analysis
|
Stages up to the Verge of Summability
Author
Dr. phil. habil. Hans-Heinrich Körle
Marburg an der Lahn, Deutschland
[email protected]
ISBN 978-3-11-034372-4
e-ISBN (PDF) 978-3-11-035983-1
e-ISBN (EPUB) 978-3-11-039916-5
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A CIP catalogue record for this book has been applied for at the Library of Congress.
Bibliografische Information der Deutschen Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data are available on the Internet at http://dnb.dnb.de.
© 2015 Walter de Gruyter GmbH, Berlin/Boston
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Printed in Germany
www.degruyter.com
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Meinen beiden Enkeln
JAKOB und MORITZ
Foreword
Imagine speedy Achilles chasing tortoises. Whenever he caught one, he couldn’t believe it.
Or rather he shouldn’t. Being a Greek, he should have thought: Every time I get where the
tortoise has been it is gone; hence, not any time I would close up. It cannot be thought – said
an authority, the philosopherZeno of Elea. A good two millenniums had to go by until a
mathematician figured out that it happens. Thanks to infinite series… (By the way: No “mil-
lennia”, for I won’t write “formulae”.)
The infinite, that means the infinitely small and, on the other hand, the infinitely large and
many. In recent history, “infinitesimals” of space, time and matter took the place of the atoms
which the ancient Greeks had puzzled over. In the end, their so-called “geometers” coped with
the infinite. They evaded it through finite procedure and thus cultivated a virus whose out-
growth was to become mathematics’ main branch: analysis. Those Greek mathematicians
were far away from the Calculus Newton and Leibniz created, way back – and ahead!
The Greeks had mastered the infinite. With Newton, it is another sort of mastery. Infinite
series were made his willing servants, i.e. series of functions which in fact had been em-
ployed before, in India. Soon they would develop a momentum of their own. Newton, Euler,
Abel, Cauchy set milestones on the long march of power series, Euler and Fourier introduced
trigonometric series. Except for Abel’s initiative, theirs needed to be clarified. Analysts like
Dirichlet, Riemann, Dedekind, Weierstraß cared to consolidate analysis such that, eventually,
one could count on infinite series: Calculus then justly applied to them and every convergent
series would definitely be worth a value, a limit number that is. We report what was accom-
plished around 1900.
Infinite series cannot be separated from general analysis. Their common history will be
outlined here along significant contributions of the pioneers. There was a prodigy among them:
Leonard Euler. Power series; to him they were the bulls to be taken by the horns. In an unortho-
dox fashion he made marvelous findings like the exponential function; in retrospect they
proved correct. Euler had considered any series “valuable”, able to be valued if only properly
formed. Indeed, divergent series were to become domesticated. By spotlighting, we will point
out the beginnings of “limitation”, thus giving a glimpse on an overall theory of infinite series.
The past may and must serve the present. Not in the least, I want to serve the needs of
those who struggle with advanced calculus. To learn, in mathematics, is to comprehend. To
learn the essentials of analysis, it pays off a lot to see how they developed in history: by trial
and error, by necessity. Infinite series prove good teachers.
Through extensive references, I tried to make background and sources available to ever-
yone. Don’t mind if you don’t find because I didn’t remind myself – like not finding out
when and where to have met the graphic which I chose for the cover’s emblem.
* * *
VIII Foreword
If you might welcome the book, then you are indebted to whom I owe it: To Professor Dr.
Thomas Sonar from Tech.Univ.Braunschweig, without whose firm encouragement I hardly
would have written it. I thank the publisher DE GRUYTER OLDENBOURG for their confi-
dence in the author and feel much obliged to Dr.Gerhard Pappert of OLDENBOURG who
gave me all, and more, support I could think of. He was engaged in the project on its first
steps. The trouble with the last ones I shared, by harsh compromise, with the busy Project
Editor, Mr. Leonardo Milla. I must thank for being granted my references. Ms. Siobhan
Scannell, M.A., attended to the language control; Mrs. Nicole Schwarz, the graphic artist,
and Mrs. Cornelia Horn did fine jobs on the figures outside and inside the book. Gratitude is
due to my home institution, the Fachbereich Mathematik und Informatik at Philipps-
Universität Marburg. I have been well equipped and enjoyed professional help from our li-
brarian, Mrs.Christa Seip, as well as from the EDP expert Mr. Burkhardt Fischer, grad.eng.,
who, ever ready for help, was welcome all the more as software is hardware to me, a red rag
to an obstinate greenhorn. For the same reason, I highly appreciated assistance by Dr.Werner
Liese: Again, I could readily produce any formula thanks to his version of LaTeX, by the
name of LiTeX.
Marburg, June 2015 Hans-Heinrich Körle
Contents
Foreword VII
1 The Greek era: Speculation, ideas and the comet of rigor 1
1.1 Infinity, challenging Greek philosophers ................................................................... 1
1.2 Infinity, mastered by a Greek “geometer”: Eudoxus .................................................. 2
1.3 Archimedes and the geometric series to be ................................................................ 5
1.4 Memorizing a joint venture ........................................................................................ 9
2 Successful experiments on infinite summation 11
2.1 Oresmus and the geometric series ............................................................................ 11
2.2 Freestyle: Cavalieri summing indivisibles ............................................................... 13
2.3 Leibniz’ telescopes ................................................................................................... 16
3 Series of functions:the prototypes 19
3.1 Preliminary: Aspects ................................................................................................ 19
3.1.1 Series versus sequences ........................................................................................... 19
3.1.2 Series: evaluation and expansion ............................................................................. 20
3.2 The power of power series ....................................................................................... 23
3.2.1 Nīlakantha and the arcustangent series .................................................................... 23
3.2.2 The hyperbolic logarithm and its series ................................................................... 26
3.2.3 Newton and the binomial series ............................................................................... 29
3.2.4 Taylor series and expansion; ordinary power series ................................................. 32
3.2.5 The binomial series in the hands of Euler ................................................................ 36
3.2.6 Euler’s algebraical analysis: evaluation and valuation ............................................. 42
3.3 The power of trigonometric series ........................................................................... 46
3.3.1 Facing antipodes ...................................................................................................... 46
3.3.2 Euler evaluating a trigonometric series .................................................................... 46
3.3.3 Priming problems ..................................................................................................... 49
3.3.4 The mysterious . . . Fourier series ............................................................................ 50
3.4 Cauchy ..................................................................................................................... 55
3.4.1 Series of powers in the hands of Cauchy and Laurent ............................................. 55
3.4.2 The Cauchy product ................................................................................................. 61
X Contents
4 Series seriously: the Greek comet reappears 67
4.1 Rigor in retrospect and prospect ............................................................................... 67
4.2 Farewell to the infinitely small; the “epsilontics” ..................................................... 69
4.2.1 Convergence,continuity ........................................................................................... 69
4.2.2 Dirichlet, Heine: uniform continuity ........................................................................ 71
4.2.3 From Abel to Weierstraß: uniform convergence ....................................................... 72
4.3 Welcome to irrationals: the complete space of real numbers .................................... 78
4.4 Another complete space: the home of convergent sequences ................................... 84
5 On the verge of summability 89
5.1 Divergent series: suspected and respected ................................................................ 89
5.2 The initiation of Cesàro and Abel summation .......................................................... 90
5.2.1 Grandi’s series, recurrent series ................................................................................ 90
5.2.2 Frobenius’ theorem and the limit theorems of Cauchy and Abel .............................. 92
5.2.3 The Cauchy product revisited by Abel and Cesàro ................................................... 94
5.2.4 Introducing the methods of basic Cesàro and of Abel means .................................. 95
5.3 Features of Cesàro and Abel means .......................................................................... 98
5.3.1 Inclusion, limitation, efficiency ................................................................................ 98
5.3.2 The C limit: a continuous functional on a Banach space ...................................... 101
1
5.3.3 An early triumph of arithmetic means: Fejér’sTheorem ........................................ 103
5.3.4 On the scale of Cesàro means ................................................................................. 103
5.3.5 Cesàro’s Cauchy product ........................................................................................ 106
5.3.6 Inverse theorems, Tauber’s theorem ....................................................................... 107
Appendixes (ofnotes, proofs, exercises) 111
Life Data 119
References 121
Index 127
